Calculating Work Done Separating Point Charges

This problem asks for the work required to move two point charges, initially separated by a specific distance, infinitely far apart. This work is directly related to the change in the electrostatic potential energy of the system.

Physics Principles Involved

• The electrostatic potential energy (U) between two point charges, $q_1$ and $q_2$, separated by a distance $r$, is given by the formula: $ U = k \frac{q_1 q_2}{r} $ where $k$ is Coulomb's constant ($k \approx 9 \times 10^9 \, \text{Nm}^2/\text{C}^2$).
• The work done (W) by an external agent to change the configuration of a system of charges from an initial state to a final state is equal to the change in the system's potential energy: $ W = \Delta U = U_{final} - U_{initial} $
• When charges are separated infinitely far apart, their final potential energy is considered to be zero ($U_{final} = 0$).

Step-by-Step Calculation

1. Identify the given values:
   • Charge $q_1 = +4 \, \mu\text{C} = +4 \times 10^{-6} \, \text{C}$
   • Charge $q_2 = -3 \, \mu\text{C} = -3 \times 10^{-6} \, \text{C}$
   • Position of $q_1$: $(-6 \, \text{cm}, 0, 0)$
   • Position of $q_2$: $(6 \, \text{cm}, 0, 0)$
   • Coulomb's constant $k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2$
2. Calculate the initial separation distance ($r$) between the charges:

   The charges are located on the x-axis at $x = -6 \, \text{cm}$ and $x = 6 \, \text{cm}$.

   $ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $ $ r = \sqrt{(6 \, \text{cm} - (-6 \, \text{cm}))^2 + (0 - 0)^2 + (0 - 0)^2} $ $ r = \sqrt{(12 \, \text{cm})^2} = 12 \, \text{cm} $ Convert the distance to meters: $ r = 12 \, \text{cm} = 0.12 \, \text{m} $
3. Calculate the initial potential energy ($U_{initial}$) of the system: $ U_{initial} = k \frac{q_1 q_2}{r} $ $ U_{initial} = (9 \times 10^9 \, \text{Nm}^2/\text{C}^2) \frac{(4 \times 10^{-6} \, \text{C})(-3 \times 10^{-6} \, \text{C})}{0.12 \, \text{m}} $ $ U_{initial} = (9 \times 10^9) \frac{-12 \times 10^{-12}}{0.12} \, \text{J} $ $ U_{initial} = (9 \times 10^9) \frac{-12 \times 10^{-12}}{12 \times 10^{-2}} \, \text{J} $ $ U_{initial} = 9 \times 10^9 \times (-1 \times 10^{-10}) \, \text{J} $ $ U_{initial} = -0.9 \, \text{J} $
4. Determine the final potential energy ($U_{final}$):

   When the charges are separated infinitely far apart, the final potential energy is zero.

   $ U_{final} = 0 \, \text{J} $
5. Calculate the work done ($W$) to separate the charges: $ W = U_{final} - U_{initial} $ $ W = 0 \, \text{J} - (-0.9 \, \text{J}) $ $ W = 0.9 \, \text{J} $

Conclusion

The amount of work required to separate the two charges infinitely far apart is 0.9 J. This positive value indicates that work must be done against the attractive electrostatic force between the opposite charges to move them infinitely far away from each other.